Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers&rdquot; to search for triangle centers.

\(a:b:c=6:9:13\).


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\(\mathbf{5c}\) \((312)\)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.242831320268&{}:{}&-0.131500522347&{}:{}&1.374331842615&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.089752100890&{}:{}&0.087131412062&{}:{}&1.002620688828&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.411869584696&{}:{}&-0.372925602060&{}:{}&1.784795186756&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.996520850439&{}:{}&0.579746034956&{}:{}&1.416774815482&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.624800511857&{}:{}&-0.563116683747&{}:{}&2.187917195604&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-19.265307708125&{}:{}&-45.596942343280&{}:{}&65.862250051405&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.561189011933&{}:{}&0.345279486078&{}:{}&1.215909525855&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.218348173783&{}:{}&-0.327522260674&{}:{}&1.545870434457&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.314377808135&{}:{}&-0.139070657065&{}:{}&1.453448465200&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.354186518322&{}:{}&-0.650372901018&{}:{}&2.004559419340&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.900234034020&{}:{}&-0.487504023687&{}:{}&2.387738057707&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.149362843508&{}:{}&0.091897604459&{}:{}&1.057465239049&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.128135977182&{}:{}&-0.303270980780&{}:{}&1.431406957962&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.295028954361&{}:{}&0.286414347270&{}:{}&1.008614607091&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-10.132653854062&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.187063003978&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.072782724594&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-19.265307708125&{}:{}&-45.596942343280&{}:{}&65.862250051405&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.561189011933&{}:{}&0.345279486078&{}:{}&1.215909525855&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.218348173783&{}:{}&-0.327522260674&{}:{}&1.545870434457&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.242831320268&{}:{}&-0.131500522347&{}:{}&1.374331842615&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.314377808135&{}:{}&-0.139070657065&{}:{}&1.453448465200&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.354186518322&{}:{}&-0.650372901018&{}:{}&2.004559419340&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.900234034020&{}:{}&-0.487504023687&{}:{}&2.387738057707&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.089752100890&{}:{}&0.087131412062&{}:{}&1.002620688828&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.149362843508&{}:{}&0.091897604459&{}:{}&1.057465239049&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.128135977182&{}:{}&-0.303270980780&{}:{}&1.431406957962&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.295028954361&{}:{}&0.286414347270&{}:{}&1.008614607091&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.411869584696&{}:{}&-0.372925602060&{}:{}&1.784795186756&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-19.265307708125&{}:{}&-45.596942343280&{}:{}&65.862250051405&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.561189011933&{}:{}&0.345279486078&{}:{}&1.215909525855&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.218348173783&{}:{}&-0.327522260674&{}:{}&1.545870434457&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.314377808135&{}:{}&-0.139070657065&{}:{}&1.453448465200&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.354186518322&{}:{}&-0.650372901018&{}:{}&2.004559419340&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.900234034020&{}:{}&-0.487504023687&{}:{}&2.387738057707&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.996520850439&{}:{}&0.579746034956&{}:{}&1.416774815482&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.624800511857&{}:{}&-0.563116683747&{}:{}&2.187917195604&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.314377808135&{}:{}&-0.139070657065&{}:{}&1.453448465200&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.354186518322&{}:{}&-0.650372901018&{}:{}&2.004559419340&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.900234034020&{}:{}&-0.487504023687&{}:{}&2.387738057707&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Hiroyasu Kamo